Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
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36 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
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52 views

Relative de Rahm cohomology computation for two disjoint circles embedded in $\mathbb{R}^2$

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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Can a volume form on a submanifold be extended to a parallel form in a neighbourhood?

Let $(M^{n+1},g)$ be a Riemannian manifold and let $\Sigma^n \hookrightarrow M$ be a smooth, closed, embedded submanifold. Let $\Omega$ be the volume form of $\Sigma$. It is well-known that a volume ...
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A manifold is a covering space over its quotient by a group action

Let $M \times G\to M$ be a properly discontinuous, free action of group $G$ on a manifold $M$. The quotient topology of the orbit space is Hausdorff. Suppose $p\in M$. How can we choose an open ...
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87 views

How “far” a differential form is from an exterior product

Consider two differential manifolds $X$ and $Y$. Consider now a differential form (of any order) $\omega$ on $X\times Y$. The easiest example is taking $\omega=\xi\wedge\eta$, where $\xi$ is a ...
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1answer
31 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
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18 views

On singular points of parallels

Say $\gamma$ is a unit speed curve and its parallel is given by $$ p (t) = \gamma (t) + d n(t)$$ where $n$ is the unit normal vector and $d$ is some scalar. I read that The parallels of a ...
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9 views

Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...
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1answer
43 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
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29 views

On reparametrisation of curves (sorry for trivial question but I'm confused)

I'm confused about speed and reparametrisations of curves. To illustrate my confusion please let me elaborate using the simplest example I could think of: Let $\gamma : [0, 2 \pi ) \to \mathbb R^2$ ...
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1answer
14 views

Parallels of a parameterised curve if not unit speed

I just read that if $\gamma$ is a curve given in unit speed parametrisation then the parametrisation of a parallel curve is given by $$ p(s) = \gamma (s) + d n(s)$$ where $n$ is the unit normal to ...
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37 views

What's book that I should read? [on hold]

When I read the chapter 4 of "Three Manifolds with Positive Ricci Curvature," I got stuck. I don't know what Fourier transform variable is, what derivative of second order nonlinear partial ...
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26 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
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1answer
64 views

Connection between harmonic functions, Bochner Laplacian and Ricci curvature

I stumbled upon the following claim in a paper: "We write the (Bochner) Laplacian in suffix notation: $\Delta_B = \nabla ^k \nabla_k$". after this statement, the following is written: ($M$ is a ...
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1answer
38 views

Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, ...
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1answer
104 views

Convex polyhedron and its Gauß-curvature

I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. What we know: Gauß-Curvature at every vertex is given by $K(p) = 2\pi - \sum\limits_{\text{angle } ...
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36 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
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21 views

Derivative group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
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2answers
81 views

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
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1answer
54 views

Statement about the isometries of a product manifold

I'm studying Minkowski spacetime $\Bbb{M}$, and I would like to make the following statement about its symmetry transformations. Since $\Bbb{M}$ is the product manifold of time and space, it inherits ...
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33 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
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1answer
40 views

Extension of vector to vector field and curvature two-form

Let $(P,\pi, M)$ be a principal bundle with structure group $G$ and let $\omega$ be a connection on this bundle. The curvature two-form is $\Omega = D\omega$ and it's quite easy to show that ...
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1answer
92 views

Some possible mistakes in Bott and Tu

I've been reading Bott and Tu's book "Differential Forms in Algebraic Topology" and it seems that this book has an enormous list of errors. Therefore I would like to confirm if the parts that I will ...
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1answer
254 views

Partition of Unity question

I am starting to read the book "Differential Forms in Algebraic Topology" by Bott and Tu. In the proof of the exactness of the Mayer - Vietoris sequence (Proposition 2.3, page 22 - 23) a partition ...
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3answers
200 views

Is a ball noncompact?

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
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29 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
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1answer
34 views

second fundamental form and connection forms

I am reading this paper that has the following: Suppose $M$ is an (n-1) dimensional closed hyper surface immersed in $\mathbb{R}^{n}$. Let $e_1, \cdots, e_n$ be orthonormal frame in $\mathbb{R}^n$ ...
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52 views

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
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Boileau-Orevkov on the intersection of a complex curve with 3-sphere

Motivation: Feel free to skip. Let $\Sigma$ be a complex curve in $\mathbb{C}^2$. If $\Sigma$ is transverse to a 3-sphere $S^3 \subset \mathbb{C}^2$ of radius $r$, then $\Sigma \cap S^3$ is a link, ...
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1answer
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Compute in the chosen charts of $M$ and $S^1$ the expression of $DF_{(5,0,-4)}$

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We ...
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1answer
242 views

A tensor calculus problem

If the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^i\lambda_i=0$ where $\lambda_i$ is a given covariant vector, show that $$a_{ij}+a_{ji}=\lambda_iv_j+\lambda_j v_i$$ where ...
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2answers
42 views

What am I doing wrong? Derivative of pedal

Let $\gamma$ be a unit speed curve $\gamma : I \to \mathbb R^2$. The pedal is given by $P (s) = (\gamma (s) \cdot N(s)) N(s)$. I tried to calculate the derivative as follows: $$ P' = (\gamma N)' N ...
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Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
4
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1answer
38 views

Parametrizations and coordinates in differential geometry - what's the difference?

From what I've read one can introduce the notion of a tangent vector to a point on a manifold in terms of an equivalence class of curves passing through that point (the equivalence relation being that ...
4
votes
1answer
53 views

Parametrizing the time an element stays in an open subset

Let $X$ be a topological space (If it helps anything, we can assume $X\subseteq\mathbb{R}^n$ or $X$ being a smooth manifold.) and $U\subseteq [0,1]\times X$ an open subset. Does there exist a ...
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1answer
38 views

Unit sphere and Ricci curvature

Why is it that on the unit sphere the Ricci curvature Ric = g (where g is the metric defined on the unit sphere) ?
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34 views

Natural derivative of Vector Fields on manifolds

I'm learning about connections and my book says that there is no natural derivative for a vector field on a manifold. Wouldn't it be possible to cook up a connection by just letting $\nabla_{v_p}X = ...
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Normal Vectors to Action of Orthogonal Group

Let $X\in\mathbb{R}^{n\times r}$ be a fixed matrix with orthogonal columns, and let $U\in\mathbb{R}^{n\times r}$ be given. Because the group of orthogonal $r\times r$ matrices, $O(r)$, is a compact ...
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1answer
29 views

Making a bijection into a diffeomorphism

Given a set $M$, one that can be made into a smooth manifold, and a bijection $f:M\to M$, does there exist a differentiable structure on $M$ such that $f$ is a diffeomorphism? In case it's not always ...
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1answer
30 views

Determine the lines of curvature of $z=xy$

I have to find the lines of curvature of $z=xy$ I calculate Weingarten Matrix as described below $p_u = (1, 0, v),p_v=(0,1,u),\nu =\frac{1}{\sqrt{1+u^2+v^2}}(-v,-u,1)$ so, $E=1+v^2,F=uv,G=1+u^2$ ...
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Nonexistence of local isometry between equidimensional Riemannian manifolds

Recall that all inner product spaces of the same dimension are isometric. For example, if $(M,\mathrm{g})$ and $(N,\mathrm{h})$ are Riemannian manifolds of the same dimension, then ...
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Boundedness of the norm of the Riemann curvature tensor

Let $(M,g)$ be a Riemannian manifold and let $R(X,Y)Z$ be its $(3,1)$ Riemann curvature tensor given by $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ Let the input vectors $X,Y,Z$ ...
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1answer
28 views

Lagrange multipliers and critical points (differential form description).

On $M \times V^*$, where $M$ is a differentiable manifold (not necessarily equipped with a metric) and $V^*$ is dual to a vector space $V$, one can define a Lagrange function $F = f +v^*h(x)$ using ...
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0answers
27 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
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1answer
55 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
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1answer
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What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles?

I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold ...
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1answer
40 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
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1answer
24 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
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1answer
39 views
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Parallel transport of a vector in hyperbolic space, specifically in $\mathbb{H}$

Let us consider Poincaré's upper plane which is defined as $\mathbb{H} = \{ (x,y) | y>0\}$. This space has a Riemannian metric $g = \text{diag}(1/y^2, 1/y^2)$. Now let us consider a differential ...